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On the convergence of some quadrature rules for Cauchy principal-value and finite-part integrals

Über die Konvergenz von gewissen Quadratursätzen für Cauchy, Hauptwert und Finite Part Integrale

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Abstract

In this paper sufficient conditions are derived to ensure the convergence of the Elliott and Hunter types of quadrature rules for the evaluation of weighted Cauchy principal-value integrals of the form:

The simultaneous convergence in the interval (−1, 1) of both quadratures was established for a class of Hölder-continuous functionsf(f∈H μ ). Corrections of some previous statements on the subject of convergence of such quadratures are also included.

Moreover, a simple derivation of the Hunter and Elliott types of quadrature rules for the evaluation of the derivative of thep-th-order of the abovestated integral was given and sufficient conditions for the convergence of the Hunter-type quadrature were obtained. Thus, the convergence of this integral was ensured for functionsf such thatf (p) ∈H μ .

Zusammenfassung

In diesem Artikel sind hinreichende Bedingungen, welche die Konvergenz von Quadratursätzen des Elliott-und Hunter-Typus für die Bestimmung von gewichteten Cauchy Hauptwert-Integralen der Form

sicherstellen, hergeleitet.

Die gleichzeitige Konvergenz beider Quadraturen im Intervall (−1, +1) wurde für eine Klasse von Hölderstetigen Funktionenf(f∈H μ ) nachgewiesen. Im Artikel sind auch Korrekturen von gewissen früheren Darlegungen über die Konvergenz von solchen Quadraturen enthalten.

Ferner wurde eine einfache Herleitung der Elliott-und Hunterschen Quadratursätze für die Bestimmung derp-ten Ableitung des obenstehenden Integrals gegeben und hinreichende Bedingungen für die Konvergenz der Hunterschen Quadratur wurden erhalten. Die Konvergenz dieses Integrals wurde somit für Funktionenf, für welchef (p) ∈H μ gilt, sichergestellt.

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Authors and Affiliations

  1. Department of Theoretical and Applied Mechanics, The National Technical University of Athens, 5, Heroes of Polytechnion Avenue Zographou, 624, Athens, Greece

    G. Tsamasphyros & P. S. Theocaris

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  1. G. Tsamasphyros
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  2. P. S. Theocaris
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Tsamasphyros, G., Theocaris, P.S. On the convergence of some quadrature rules for Cauchy principal-value and finite-part integrals. Computing 31, 105–114 (1983). https://doi.org/10.1007/BF02259907

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  • DOI: https://doi.org/10.1007/BF02259907

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